alg_st_notes_3_annotated.pdf - THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1231 MATHEMATICS 1B ALGEBRA Section 3

alg_st_notes_3_annotated.pdf - THE UNIVERSITY OF NEW SOUTH...

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THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1231 MATHEMATICS 1B ALGEBRA. Section 3: - Eigenvectors and Eigenvalues. 1. Motivation and De fi nitions Consider the e ff ect of multiplying vectors in R 2 by the matrix A = ± 1 2 1 0 ² . For example A ± 1 1 ² = ± 3 1 ² and A ± 2 1 ² = ± 4 2 ² 0 1 2 3 0 1 2 3 x A x 0 1 2 3 4 0 1 2 3 x A x In the fi rst case, we see that the vector ± 1 1 ² gets stretched and rotated, but in the second case, the vector ± 2 1 ² just gets ‘stretched’ by a factor of 2. Now look what A does to the vectors ± 1 0 ² and ± 1 1 ² . A ± 1 0 ² = ± 1 1 ² and A ± 1 1 ² = ± 1 1 ² . x A x A x x Once again we see that ± 1 0 ² is moved around, while the image of the ± 1 1 ² is parallel and of the same length, but in the opposite direction. We can think of this as a ‘stretch’ by 1
the factor 1. The vectors that got ‘stretched’ (possibly in a negative sense) are called the eigenvectors for the matrix A and the ‘stretch factor’ (in the above examples they were 2 and 1 ) are called the eigenvalues of the matrix. These eigenvectors and eigenvalues play an extremely crucial role throughout many areas of mathematics and its applications in Engineering and Physics. You might like to compare this with what happens when we di ff erentiate e λ x . The derivative is λ e λ x . In other words, when we di ff erentiate e λ x , this function gets ‘stretched’ by a factor of λ . We will exploit this idea later when looking at systems of di ff erential equations. In the meantime, let us try to gather together what we have done and see what the implications are.

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